NCHRP Web-Only Document 199: Estimating Joint Probabilities of Design Coincident Flows at Stream Confluences Roger T. Kilgore David B. Thompson David T. Ford Kilgore Consulting and Management Denver, CO Contractor’s Final Report for NCHRP Project 15-36 Submitted March 2013 National Cooperative Highway Research Program ACKNOWLEDGMENT This work was sponsored by the American Association of State Highway and Transportation Officials (AASHTO), in cooperation with the Federal Highway Administration, and was conducted in the National Cooperative Highway Research Program (NCHRP), which is administered by the Transportation Research Board (TRB) of the National Academies. COPYRIGHT INFORMATION Authors herein are responsible for the authenticity of their materials and for obtaining written permissions from publishers or persons who own the copyright to any previously published or copyrighted material used herein. Cooperative Research Programs (CRP) grants permission to reproduce material in this publication for classroom and not-for-profit purposes. Permission is given with the understanding that none of the material will be used to imply TRB, AASHTO, FAA, FHWA, FMCSA, FTA, Transit Development Corporation, or AOC endorsement of a particular product, method, or practice. It is expected that those reproducing the material in this document for educational and not-for-profit uses will give appropriate acknowledgment of the source of any reprinted or reproduced material. For other uses of the material, request permission from CRP. DISCLAIMER The opinions and conclusions expressed or implied in this report are those of the researchers who performed the research. They are not necessarily those of the Transportation Research Board, the National Research Council, or the program sponsors. The information contained in this document was taken directly from the submission of the author(s). This material has not been edited by TRB. The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. On the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Ralph J. Cicerone is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Charles M. Vest is president of the National Academy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, on its own initiative, to identify issues of medical care, research, and education. Dr. Harvey V. Fineberg is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy’s purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Ralph J. Cicerone and Dr. Charles M. Vest are chair and vice chair, respectively, of the National Research Council. The Transportation Research Board is one of six major divisions of the National Research Council. The mission of the Transporta- tion Research Board is to provide leadership in transportation innovation and progress through research and information exchange, conducted within a setting that is objective, interdisciplinary, and multimodal. The Board’s varied activities annually engage about 7,000 engineers, scientists, and other transportation researchers and practitioners from the public and private sectors and academia, all of whom contribute their expertise in the public interest. The program is supported by state transportation departments, federal agencies including the component administrations of the U.S. Department of Transportation, and other organizations and individu- als interested in the development of transportation. www.TRB.org www.national-academies.org AUTHOR ACKNOWLEDGMENTS The research reported herein was performed under NCHRP Project 15-36 by Kilgore Consulting and Management. Mr. Roger T. Kilgore was the Project Director and co-Principal Investigator. The co-authors for this report are Dr. David B. Thompson with R.O. Anderson, Inc. and Dr. David Ford of Ford Consulting Engineers, Inc. Additional support was provided by Brad Moore, P.E. and Nathan Pingel of David Ford Consulting Engineers, Inc. Mark Heggli of Innovative Hydrology and John Henz of HDR, Inc. provided literature review and analysis of the nature of storm cells and radar as potential tools for assessment. ABSTRACT The results of a research study to develop practical procedures for estimating joint probabilities of coincident flows at stream confluences and guidelines for applying the procedures to design problems are documented in this report. The study was based on a database of 85 gage pairs located throughout the coterminous United States. Two practical design methods emerged from the study. The first method for determining a set of exceedance probability combinations associated with the desired joint probability for design is based on copulas. The Gumbel- Hougaard copula was selected as the most appropriate for this application. Use of the copula requires estimation of the Kendall’s τ rank correlation for an ungaged watershed pair. Tools for estimating this parameter based on watershed characteristics are also a product of the research. For a given Kendall’s τ, the appropriate combinations of events on the main and tributary streams for a desired joint exceedance probability are provided in tabular form. The second method is the use of the total probability method. Although the total probability method, based on the total probability theorem, is a well-documented procedure, this study provided recommended conditional probability matrices needed to apply the theorem. Both methods result in the design conditions representing the joint probability desired by the designer for analysis. Table of Contents List of Tables ................................................................................................................ iv List of Figures .............................................................................................................. vi Executive Summary .................................................................................................... vii 1. Background ............................................................................................................... 1 1.1 Problem Statement ........................................................................................... 1 1.2 Literature Review .............................................................................................. 3 1.3 Research Objectives and Scope ....................................................................... 4 2. Research Method ...................................................................................................... 5 2.1 Work Plan ......................................................................................................... 5 2.1.1 Phase I ....................................................................................................... 5 2.1.1.1 Task 1. Critically Review Literature/Summarize Procedures ........................... 5 2.1.1.2 Task 2. Critically Review Procedures .............................................................. 5 2.1.1.3 Task 3. Identify Confluent Stream Sites .......................................................... 5 2.1.1.4 Task 4. Propose Procedures ........................................................................... 5 2.1.1.5 Task 5. Submit Interim Report ......................................................................... 6 2.1.2 Phase II ...................................................................................................... 6 2.1.2.1 Task 6. Collect/Compile Data .......................................................................... 6 2.1.2.2 Task 7. Develop Practical Procedures............................................................. 6 2.1.2.3 Task 8. Validate Procedures ........................................................................... 6 2.1.2.4 Task 9. Final Report .......................................................................................